You Get What You Share: Incentives for a Sharing Economy

The Thirty-Third AAAI Conference on Artiﬁcial Intelligence (AAAI-19)

Sreenivas Gollapudi Kostas Kollias Debmalya Panigrahi Google Research Google Research Duke University

Abstract their individual resources. The platform allows the agents to In recent years, a range of online applications have facilitated list and search for available resources, which enables them resource sharing among users, resulting in a significant in- to identify partners and form sharing groups. More specif- crease in resource utilization. In all such applications, shar- ically, in workforce and educational applications, there are ing one’s resources or skills with other agents increases so- groups that work to complete a task or a project, while in cial welfare. In general, each agent will look for other agents ride sharing and room sharing applications, the notion of a whose available resources complement hers, thereby form- group appears when agents get together to use a provided ing natural sharing groups. In this paper, we study settings resource, e.g., a ride or a house. where a large population self-organizes into sharing groups. A natural goal is to partition the users into sharing groups In many cases, centralized optimization approaches for creat- ing an optimal partition of the user population are infeasible that maximize the overall utility or social welfare of the because either the central authority does not have the neces- system. One may model this as an optimization problem, sary information to compute an optimal partition, or it does where a centralized authority computes the optimal parti- not have the power to enforce a partition. Instead, the central tion of users. However, in typical environments, the informa- authority puts in place an incentive structure in the form of tion necessary for computing this partition is hard to obtain a utility sharing method, before letting the participants form due to privacy issues or due to the sheer size of the popu- the sharing groups by themselves. We first analyze a simple lation. Moreover, the central authority has only limited con- equal-sharing method, which is the one most typically en- trol over the participants, and often cannot enforce the parti- countered in practice and show that it can lead to highly inef- tion. Hence, in this paper, we study the following question: ficient equilibria. We then propose a Shapley-sharing method Consider a population of individuals, who can form shar- and show that it significantly improves overall social welfare. ing groups and share their individual resources with group members. What is the right way to distribute the welfare pro- Introduction duced by each sharing group among its members so that the The sharing economy (Belk 2014) is a term used to describe population self-organizes in a socially optimal manner? The the modern trend of using online platforms and applications semantics of distributing welfare are different for each ap- to increase the value of certain resources (such as goods, ser- plication. For instance, in online learning, utility is given vices, or skills) by enabling the sharing and reuse of these in the form of course credits that depend on how students resources. Carpooling and ride sharing applications (e.g., contributed to group projects. In the ride-sharing applica- Uber, Lyft, RideWith) allow people to better utilize com- tion, using a vehicle has some value that is the same for muting resources (cars, fuel, etc.), lodging and room sharing all agents and monetary transfers reward those providing re- applications (e.g., AirBnB) allow users to share their space sources and extract payments from those utilizing resources when not using it, online learning platforms (e.g., Coursera, to adjust utilities accordingly. Udacity, edX) allow students to share knowledge and col- We model instances of this problem as games that we call lectively learn, and workforce applications (e.g., Upwork, resource sharing games. The most natural way to split the Amazon Mechanical Turk) allow workers to collaboratively produced welfare among the group’s members is to divide it complete requested projects by contributing complementary equally. In fact, this is the method that naturally comes into skills. In all such applications, there are natural limits on the play when there are no monetary or credit transfers in our extent to which the resources can be shared. In carpooling, applications of interest, i.e., everyone simply gets access to each car can fit a small number of people; in lodging, each the resources available in the group. However, we show that room or house has a fixed capacity; in crowsourced projects this method can perform quite sub-optimally with respect to and online courses, there is a prescribed size for working overall social welfare. We then propose an incentive struc- groups. In effect, this means that the agents need to be or- ture derived from the well-known Shapley value in the eco- ganized in groups of limited size so as to effectively share nomics literature (Shapley 1953), and show that the quality Copyright c 2019, Association for the Advancement of Artificial of the resulting outcomes is significantly improved. Intelligence (www.aaai.org). All rights reserved. In our model, each participant owns a subset of all possi-

2004 ble resource types and, when a group is formed, everyone in (in the form of a utility-sharing method) that will lead the the group has access to all resource types that at least one of agents to efficient outcomes. them owns. An important observation is that the resulting so- We begin by studying the most natural and widely used cial welfare function is not submodular, but supermodular. utility sharing method, which dictates that each agent has (This is in contrast to an existing body of work that studies utility equal to the number of resources available in her utility games for submodular welfare functions, e.g., (Vetta sharing group. This is equivalent to equally splitting the so- 2002; Gairing 2009; Marden and Wierman 2013).) Infor- cial welfare produced by a group among its members and, mally, this means that a group produces more utility than hence, we call it equal-sharing (EQUAL). For instance, in the sum of its individuals. While this is often the case in online course projects, EQUAL would mean all students in a practice, it can be theoretically problematic because the only group will get the same course credit, i.e., the problems that stable solution is typically one where all the agents join the the group has solved. In the carpooling application, EQUAL same group. However, in real-world applications (we give means all agents get to take the ride and split any costs some examples below), it is impractical to have very large (e.g., fuel) equally. We show that, for any reasonable equi- groups. So, we will adopt a model where groups are subject librium concept, there is an equilibrium partition of agents to a cardinality constraint, and also show that it is possible into sharing groups that is k times worse than the optimal. to enforce such a constraint through incentives alone. We also explain that this inefficiency with respect to the optimal is the worst possible for any partition, independent of Applications whether it is an equilibrium or not. In light of this obser- First, we describe two concrete real world examples of our vation, we seek an alternative utility-sharing method, which resource sharing games: results in better equilibria. The method we propose is derived from the Shapley value (Shapley 1953) and, hence, we • Massive Open Online Courses – Suppose we are inter- call it Shapley-sharing (SHAPLEY). SHAPLEY aims to im- ested in partitioning the set of students into groups to prove on the main shortcoming of EQUAL, which is that it complete an assignment comprising multiple problems. rewards agents based on their contribution to improving the From a social perspective, we would prefer a partition of social welfare by sharing their resources. Below we present students such that every group in the partition completes our results for the various equilibrium concepts, which show as many problems as possible. Each student on the other that SHAPLEY leads to better equilibria than EQUAL. hand, wishes to maximize the course credit she will re- Nash equilibrium. The Nash equilibrium (NE) is the most ceive. The goal of the central authority (teacher) is to pro- established equilibrium concept in the literature. An NE in vide the right incentives (in the form of a grading scheme) our setting is a partition such that no agent can improve to the students so that effective groups grow organically her utility by unilaterally deviating to a different group. It and in a decentralized fashion. is clear that the set of equilibria of a resource sharing game • Carpooling – In a carpooling system, agents form groups depends on the utility-sharing method being used. The price and repeatedly share rides to and from work during rush of anarchy (POA) (Koutsoupias and Papadimitriou 2009) is hour. In this setting the resources of the game correspond used as a metric that quantifies the performance of a utility- to car rides from location to location and welfare is gen- sharing method by calculating the worst-case ratio of the so- erated for each agent who takes a ride. From a social per- cial welfare in the optimal partition to the social welfare in spective, we wish that the commuting resources are put an NE. As we mentioned earlier, the POA of EQUAL is k. to good use. On the other hand, agents wish to optimize Unfortunately, the POA remains Θ(k) for SHAPLEY as well. their individual rewards. Providing the appropriate incen- In fact, we show that this will be the case for any natural tives and inducing the organic formation of efficient shar- utility-sharing method. However, we observe that the exam- ing groups is important in achieving the social objective. ples that yield this universal POA lower bound are unnatural as they assume agents will form groups of very small sizes, Several other applications such as crowdsourcing and labor even when they are allowed to form larger sharing groups markets similarly fit our proposed model. with more resources available. To get past these unnatural pathological cases, we consider the following two equilib- Our Contributions rium concepts. We consider a setting where a population of agents seeks to Strong Nash equilibrium. The strong Nash equilibrium share resources in groups of size at most k. The value of k, (SNE) is an NE that is robust to deviations by coalitions of the desired group size, is application specific (e.g., in (Davis agents (Aumann 1959). This implies that a partition is an 2009), values of 4, 5, and 6 are prescribed as appropriate SNE when there is no subset of agents who can coordinate for educational applications, while similar such constraints and deviate to other groups (or start groups on their own), are natural in carpooling and other applications). Each group thereby improving their individual utilities. The strong price generates social welfare, which is then distributed as util- of anarchy (SPOA) is the direct analog of the POA for the ity to the agents who generated it. We study this setting as SNE. We prove that SHAPLEY results in a significant im- a game, where the agents self-organize into groups. Each provement of the SPOA, which is brought down to 2 (from k agent wishes to optimize her own utility share. We assume for EQUAL). Unfortunately, this equilibrium concept has the the role of the central authority, who cannot enforce the final well-known drawback that it is not guaranteed to exist in sharing groups, but who can establish an incentive structure many settings. Indeed, we give examples of resource shar-

2005 ing games in our setting that have no SNE under SHAPLEY. Each agent i in group G, on the other hand, receives an indi- Toward the goal of characterizing equilibria with more de- vidual utility ui(G). Utility-sharing is budget-balanced, i.e., sirable outcomes in the context of unilateral deviations, we focus on an equilibrium concept that we define here, and ∑ ui(G) = U(G). which we argue is the one that best predicts the behavior i∈G of selfish agents in our settings. We call this new concept a We now define the equilibrium concepts that we will balanced Nash equilibrium. study. Balanced Nash equilibrium. We define a balanced Nash Definition 1. A partition is a Nash equilibrium (NE) when equilibrium (BNE) as an NE such that all groups, except pos- G for every agent i in group G, it is the case that ui(G) ≥ sibly one, have k agents, thereby ensuring balance in terms 0 0 of their size. In many applications (such as in carpooling ui(G ∪ {i}) for every other group G ∈ G . In other words, and online courses), this is an end in itself. In other set- no agent can improve her utility by switching to any other tings, minimizing the number of groups subject to cardi- sharing group unilaterally. nality constraints is a desirable outcome. Note that merg- As mentioned earlier, we also use the coalition-resistant ing small groups can only increase social welfare as more notion of strong Nash equilibrium. resources are added to the group and more agents benefit Definition 2. A partition G is a strong Nash equilibrium BNE from them. We will also show that the has the follow- (SNE) if no coalition of agents S ⊆ N can deviate (move to ing properties: another group, including forming a new group) to form a 0 0 • Ease of enforcement – It is easy for the central authority new partition G such that ui(G ) > ui(G) for every agent to slightly modify the SHAPLEY method so that every NE i ∈ S, where G0 and G are the respective groups of agent i of the resulting game is a BNE of the original game, i.e., a in G 0 and G . Intuitively, an SNE is a partition such that no BNE can be easily enforced. coalition of agents can coordinate and move between groups in a way that benefits every member of the coalition. • Existence – A BNE is always guaranteed to exist. Finally we introduce the notion of a balanced Nash equi- • Efficiency – The loss of efficiency in a BNE (we call this librium (BNE). the balanced price of anarchy (BPOA) by analogy) is at most 3 in the SHAPLEY scheme (as against the BPOA of k Definition 3. A partition is a balanced Nash equilibrium of EQUAL). (BNE) if it is an NE such that all groups, except possibly one, have k agents. Finally, we perform simulations which show that the SHAPLEY method improves on EQUAL in realistic settings, We will show existence of this equilibrium and demon- with respect to three metrics that measure our desiderata for strate that it can be easily enforced by the central author- forming sharing groups: performance (measured by the so- ity by modifying the utility-sharing method. We also note cial welfare of the partition), participation (measured by the that BNE outcomes are desirable since they enforce groups number of sharing groups of size k), and fairness (measured of large size, where sharing is maximized and social welfare by the quality of the weakest group). is increased. Having introduced our three equilibrium concepts, we are Preliminaries ready to define the metrics that we use to measure the performance of a utility-sharing method. In this section, we describe our model and introduce the notation. There is a set of agents N and a set of resource Definition 4. The price of anarchy (POA), the strong price of types P. Let Pi ⊆ P denote the set of resource types that anarchy (SPOA), and the balanced price of anarchy (BPOA) agent i ∈ N possesses. Agents self-organize into groups so are the worst case ratios of the optimal social welfare to the as to share their available resources. We denote the set of social welfare of an NE, SNE, and BNE respectively, i.e., groups by G . The social welfare of a group G ∈ G is: maxG U(G ) ( S POA = . | i∈G Pi| · |G|, if |G| ≤ k minG is an NE U(G ) U(G) = (1) maxG U(G ) 0, otherwise SPOA = . minG is an SNE U(G ) k cardinality parameter where is the that represents the max- maxG U(G ) imum group size. The expression encodes the fact that every BPOA = . min BNE U(G ) agent in a group benefits from the presence of each resource G is a available in the group. Note that this definition assumes that Equal Sharing all resources have unit value. This assumption is without loss of generality since resources with non-uniform utility can be The most natural way of sharing utility from resources in modeled by multiple resources with equal utility. The over- a group is to have every agent in the group receive utility all objective is to form sharing groups that maximize social equal to the value of the resources. Note that this is equiva- welfare lent to equally splitting the welfare generated by the group among its participants and, hence, we call this utility-sharing U(G ) = U(G). ∑ method equal-sharing (EQUAL). G∈G

2006 Theorem 5. The POA, SPOA, and BPOA are all k under • If agent i owns a resource of type p, then she gets 1 unit EQUAL. of utility for p. • If agent i does not own a resource of type p and the group Proof. Consider the following setting. There is only one has n agents who have that resource and share it with her, resource type. There are k agents who own the resource then she gets n units of utility for that resource. The re- (we call them strong agents) and k · (k − 1) agents who do n+1 ∗ 1 not (we call tham weak agents). The optimal solution G maining n+1 units are shared uniformly among those who 1 places every strong agent in a different group and adds k −1 shared resource p with her (so each of them gets n(n+1) weak agents to each group. Then all agents have access to extra utility). the unique resource type in the game and the social wel- ∗ 2 • If the size of the group is larger than k, each agent is pe- fare is U(G ) = k . Now consider the partition G where all nalized for a unit of utility per resource type. strong agents are in the same group and all other agents are arbitrarily partitioned into groups of size k. We will show SHAPLEY is connected to the the standard definition of that this partition is an SNE, an NE, and a BNE. We first argue the Shapley value. The standard definition prescribes that that it is an SNE: Note that all strong agents receive the max- each agent is rewarded with the expected increase she causes imum possible utility of 1, and hence they have no incentive to the group welfare function U(·) over a uniformly random to deviate from this partition in any way. Furthermore, the order of arrival of the agents in the group. In our setting weak agents have no way to improve their utility above 0, the exact Shapley value we get when plugging in U(·) as since if any subset of them were to join the strong group, its the welfare functions is hard to work with and get a closed cardinality would increase beyond k leading to total utility form expression. Our SHAPLEY instead is equivalent to the Shapley value under the following modification: Agents get of 0. This proves that G is an SNE. It follows that it is an NE as well, since every SNE is also an NE. Observing that utility equal to the expected increase they cause to the wel- the number of groups is equal to the ratio of the number of fare function after removing the capacity constraint (i.e. the agents and the cardinality parameter, i.e., k, proves that G is Shapley value with infinite capacity), but they also share a penalty equal to their cardinality if the size of the group also a BNE. Note that U(G ) = k. Then we get that the POA, k the SPOA, and the BPOA are all at least k. is larger than . This penalty is split evenly across agents It is not hard to see that this is actually the worst possible since they all contribute equally to bringing the group size ratio for any possible partition (not only for equilibria). For to larger than k. each resource an agent owns, the least social welfare earned SHAPLEY is an instantiation of the Shapley-value for the by that resource is 1, in cases when only that agent receives modified setting described above and, hence, guarantees the utility for it. The maximum possible is k, in cases when she existence of an NE in every resource sharing game (Kol- shares it with another k − 1 agents in her group. lias and Roughgarden 2011). Intuitively, SHAPLEY is en- gineered to address the drawbacks that caused EQUAL to We note that this strong lower bound on the SPOA sug- be very inefficient. It incentivizes strong agents to move to gests that such inefficient equilibria can appear in many set- groups where they can offer more help and also incentivizes tings of interest, with weaker notions of stability. This in- them to spread over multiple groups (since the bonus of an cludes situations where two agents can swap their group agent who shares is inversely proportional to the number of memberships or where a small number of agents can coor- owners of the same resource type in the group). At the same dinate and deviate to a new group. time, it breaks symmetry in utility earned by the agents, al- lowing them to break away from very inefficient partitions Shapley Sharing that have groups of size k. In light of the poor performance of EQUAL in terms of the Nash Equilibria three inefficiency metrics, we seek a utility-sharing method By equivalence to Shapley value, we get that an NE is that addresses the main shortcoming of EQUAL, which is that agents who significantly increase social welfare by shar- guaranteed to exist in our game for SHAPLEY utility sharing with their peers are not rewarded for it. The Shapley ing (Kollias and Roughgarden 2011). Indeed, we show in value (Shapley 1953) from the economics literature provides our full version (Gollapudi, Kollias, and Panigrahi 2018) a method for splitting social welfare among those generat- that better-response dynamics always converges to an NE. ing it. We propose applying the Shapley value in our setting, However, it turns out that any utility-sharing method, in- which yields a natural way of partially transferring utility cluding SHAPLEY, that has an agent split her utility with from the agents who do not have a resource to the agent shar- those who shared resources with her has a POA of Θ(k). ing it with them. In this section, we analyze the performance We call such methods share-rewarding. Due to space limi- tations, we omit the proof, which appears in our full version of Shapley-sharing (SHAPLEY) and show that it results in a significant improvement in the quality of equilibria. (Gollapudi, Kollias, and Panigrahi 2018). The main idea is that when an agent is allowed access to a Theorem 7. The POA of any share-rewarding method is resource type that she does not possess, she shares her 1 unit Θ(k). of utility with agents who shared the resource with her. SHAPLEY is clearly a share-rewarding method, and hence Definition 6 (SHAPLEY). SHAPLEY is based on the follow- it also has a POA of Θ(k). ing simple utility granting rules: Corollary 8. The POA of SHAPLEY is Θ(k).

2007 Strong Nash Equilibria While this bound shows that SHAPLEY outperforms In this section, we prove that the SPOA drops from k to 2 EQUAL for the SPOA metric, an SNE may not exist in all when we use SHAPLEY instead of EQUAL. games – we give such an example in our full version (Golla- pudi, Kollias, and Panigrahi 2018). Theorem 9. The SPOA of SHAPLEY is 2. Fact 10. There exist games that do not have an SNE under Proof. We prove an upper bound of 2 on the SPOA of SHAP- SHAPLEY sharing. LEY here. In our full version (Gollapudi, Kollias, and Pan- igrahi 2018), we show that this analysis is tight, i.e., there Balanced Nash Equilibria is an example of a game where the SPOA of SHAPLEY is exactly 2. We have previously seen that SHAPLEY outperforms Let G ∗ be an optimal partition of agents into sharing EQUAL for an SNE if it exists, but an SNE may not ex- groups, such that every group has at most k agents. Triv- ist in some games. On the other hand, an NE exists in ially one such optimal partition exists by the fact that any all games, but all natural utility sharing methods (share- group with more than k agents produces 0 welfare by defi- rewarding methods) are highly inefficient in this case. So, nition and, hence, breaking it into smaller groups can only the question arises: is there an equilibrium concept that can improve the social welfare. Also, let G be an SNE. Consider be guaranteed to exist and has efficiency comparable to SNE? any group G∗ of the optimal partition. We will show that the We show that a BNE achieves these properties. l m total utility of the agents from G∗ in partition G , for a given |N| Recall that a BNE is an NE with k groups, such that resource type p, is at least half the social welfare derived all groups, except possibly one, have exactly k agents each, from this resource type in G∗. where N is the set of agents. We note that the central author- Partition G is an SNE, which implies there is at least one l m ∗ |N| agent i1 from group G , whose utility in her group in par- ity can easily enforce such a partition, by prescribing k tition G is at least her utility in group G∗. Otherwise, these sharing groups that the agents should join and by assign- agents would abandon their groups in partition G and form ing negative utility to the agents not participating in the pre- ∗ ∗ ∗ G . If we remove agent i1 from G , we get group G \{i1}. scribed groups. Moreover, the authority needs to enforce a ∗ As before, there is at least one agent i2 ∈ G \{i1} whose j |N| k ∗ cardinality constraint of k for the first k groups and a utility in partition G is at least her utility in group G \{i1} ∗ j |N| k because otherwise the agents in G \{i1} would deviate to cardinality constraint of |N| − k for the last group. We ∗ k form G \{i1}. Continue in the same fashion until we have ∗ show that with SHAPLEY, we can bring the BPOA from k a complete ordering i ,i ,...,i ∗ of G , such that the util- 1 2 |G | (in EQUAL) down to at most 3. Our BPOA bound can be ity of i in her group in G is at least her utility in group l read both as a BPOA bound for the original game and as a {il,il+1,...,i|G∗|}. ∗ POA bound for the modified game that fixes the number of Let u be the sum over all l = 1,2,...,|G |, of the utility sharing groups by giving negative utility to non-prescribed of agent il, in group {il,il+1,...,i|G∗|}. As argued, this sum groups. u lower bounds the total utility of these agents in G . Let p A BNE in the original game is guaranteed to exist, as ex- denote the contribution of a resource type p to the value of hibited by the following theorem, which we prove in our full u. Now focus on an agent il and examine the utility she gets version (Gollapudi, Kollias, and Panigrahi 2018). from resource type p. There are three cases: Theorem 11. SHAPLEY always induces games such that a • If she owns p and there is also some il0 who owns p, with BNE exists. 0 l > l, then the contribution of il to up is at least 1, since she gets 1 unit of utility for owning p and we have as- We now prove our upper bound on the BPOA. The intu- sumed |G∗| ≤ k. ition behind the fact that the BPOA improves significantly, in contrast to the POA, is that almost all groups have a large • If she does not own p, but there is some il0 who owns p, number of agents and there is an incentive for strong agents with l0 > l, then the contribution of i to u is at least 1 , l p 2 to move to a group that has few resource types available and since she gets to use it once il0 shares it, in which case she 1 receive credit for sharing resources. gets at least 2 utility. ∗ Theorem 12. The BPOA of SHAPLEY is at most 3. • If il owns p, and she is the last one in our ordering of G 1 who does, then her contribution to up is at least 1 plus 2 Proof. For simplicity of exposition, in this proof we assume for every agent after her in the ordering, since she gets that the number of agents is a multiple of k. Our arguments 1 1 unit of utility for owning p and 2 for every agent she easily extend to the general case. |N| shares it with in {il,...,i|G∗|}. Let G be a BNE partition with g = k sharing groups. For the purposes of this proof we define a new notion that we From the above, it follows that we get at least 1 in u for 2 p call the uniform fractional deviation of an agent from . In every agent in G∗. The optimal partition extracts 1 unit of G such a deviation, the agent splits herself into g fractions of social welfare from each agent in G∗ for p. Summing over size 1 each and allocates one such fraction to each sharing all resource types and over all groups of G ∗, and using the g fact that the SHAPLEY method distributes precisely the so- group. When an agent picks this strategy, her utility is her av- ui(G∪{i}) cial welfare as utility, we get the SPOA upper bound of 2. erage utility across all groups, i.e., ∑G∈G g , ui(G) be-

2008 100 1,000 10 Shapley 80 950 equal

60 900 5

Shapley # full groups Shapley

Social welfare 850 40 equal equal Minimum utility 800 0 2 3 4 5 6 2 3 4 5 6 2 3 4 5 6 # resources avg agent has # resources avg agent has # resources avg agent has (a) Performance (b) Participation (c) Fairness

Figure 1: SHAPLEY yields ∼ 10% improvement of the social welfare across agent, consistently ﬁlls all groups, and signiﬁcantly improves the worst agent.

15 80 Shapley 1,000 equal 10 60 980 5

Shapley # full groups Shapley Social welfare 40 equal equal Minimum utility 960 0 600 800 1,000 1,200 1,400 1,600 600 800 1,000 1,200 1,400 1,600 600 800 1,000 1,200 1,400 1,600 # agents owning a resource type # agents owning a resource type # agents owning a resource type (a) Performance (b) Participation (c) Fairness

Figure 2: SHAPLEY yields ∼ 10% improvement of the social welfare across resource type frequency levels, consistently fills all sharing groups, and significantly improves the minimum utility. ing the utility of i in group G. When all agents play this strat- summing over all resource types would give us (2). Given egy, we get the uniform fractional partition ∗. The welfare 1 G the above, we will show that λ = µ = 2 satisfy the inequality of the uniform fractional partition is given as follows: Sup- on a per resource type basis, thus proving an upper bound of pose np agents own a resource of type p. Then there is a µ+1 λ = 3 on the BPOA. cumulative of np agents who own this resource type in each g ∗ ∗ group, since each group has a total cumulative fraction of k Suppose G yields social welfare Up(G ) as far as resource type p is concerned and that yields social welfare agents. The welfare extracted from resource type p in G ∗ G U ( ) = γU ( ∗). We assume γ ∈ [0,1], since, if this is not n np o p G p G is min 1, g kg = min gk,npk , which is the best possi- the case, we can discard this resource type from our com- ble. parison between G and G ∗, without loss of generality. There ∗ We now prove that the total welfare in the uniform frac- are |N|−γUp(G ) agents that do not own resource type p in tional partition G ∗ is at most 3 times the welfare of the G . Hence, when an agent who owns the corresponding re- BNE G . We do so by applying the (λ, µ)-smoothness frame- source type deviates to her uniform fractional strategy, she ∗ work (Roughgarden 2009) which prescribes that, if we find k |N|−γUp(G ) k will get utility of at least |N| 2 , since |N| is the size numbers λ and µ such that 1 of the fraction she assigns to each sharing group and 2 is ui(G ∪ {i}) ∗ ∑ ∑ ≥ λU(G ∗) − µU(G ), (2) the reward she gets for helping the |N| − γUp(G ) integral i∈N g agents who do not own resource type p. There are at least G∈G ∗ Up(G ) ∗ µ+1 agents who own resource type p, given that G has then the BPOA is at most . We may see this as follows: k λ U ( ∗) p By the fact that G is a BNE, we get that each agent i does social welfare p G for . Then we get that the part of the not benefit by switching to any sharing group other than her left-hand side of (2) that corresponds to p is current one. Hence, i also does not benefit by performing the uniform fractional deviation, since the resulting utility is ∗ ∗ k |N| − γUp(G ) Up(G ) the average of all possible integral deviations. Then, it fol- . ∗ |N| 2 k lows that, U(G ) ≥ λU(G ) − µU(G ), which, by rearrang- µ+1 ing, gives a λ upper bound on the BPOA. ∗ Clearly, it suffices to find λ and µ such that inequality (2) If we substitute |N| with its smaller or equal number Up(G ) ∗ is true per resource type, i.e., if the agent’s deviation utility Up(G ) we get that the above is at most 2 (1−γ), for which the on the left hand side only takes a given resource type p into ∗ Up(G ) ∗ ∗ consideration and the social welfare on the right hand side, inequality 2 (1 − γ) ≥ λUp(G ) − µγUp(G ) always 1 again only takes resource type p into consideration. Then, holds with λ = µ = 2 . This completes the proof.

2009 Simulations in Figure 1c. In contrast, SHAPLEY performs better on this In this section, we experimentally evaluate the performance metric by rewarding agents for sharing resources. of SHAPLEY vs EQUAL. In our setup, we begin with |N| = Skewed Resource Type Frequency. For our second simu- 5,000 agents and a hypothetical set of |P| = 20 resource lation, we assume that the agents are symmetric with respect types that all agents can utilize in groups of size at most to their resource ownage expressed as a fraction of resources k = 5. In each simulation, we begin with each agent in a shar- owned. Further, we assume that the frequency of each re- ing group by herself. Then, in each round we let the agents source type, defined as the number of agents who own the deviate (one by one) to a sharing group that they prefer until corresponding resource type, is drawn from a power law dis- no such deviations are possible, i.e., until an NE is reached. tribution. The plots in Figures 2a, 2b, and 2c show how our We use three metrics to evaluate the quality of the parti- three metrics vary as the average resource type frequency tion reached by the agents. The first one is the natural per- changes (i.e., as the degree of the power law changes). formance metric, which measures the social welfare of the Similar to the previous simulation, we observe in Fig- resulting partition. The second one is a participation met- ure 2b that SHAPLEY results in balanced partitions. From ric, which counts the number of sharing groups for which the plot of Figure 2a, we conclude that SHAPLEY offers a the cardinality constraint is tight. As we will see, SHAPLEY significant improvement in the social welfare for this setting almost always yields partitions such that all groups are full, as well. These two facts are further validation of our theo- which also validates our study of the BNE as an important retical analyses. equilibrium concept in this setting. Finally, we focus on a In summary, SHAPLEY outperforms EQUAL on all three fairness metric, which measures the minimum social welfare metrics. This is expected, given that now the agents are con- extracted by an agent, i.e., the number of resources available sidered symmetric and there are only a few agents who do in the weakest group. not own any resource types . This also results in the mini- In the following sets of experiments we use skewed distri- mum utility to be above 0 and the number of filled sharing butions that follow power laws. This is to model the fact that groups to increase. in most applications there is a small fraction of agents with many resources and most agents have few resources (e.g., Related Work in ride sharing settings most agents contribute a single car, with few having more than that and a tiny fraction operat- The price of anarchy was defined in (Koutsoupias and Pa- ing a fleet of cars). Similarly, there are a few types of re- padimitriou 2009) (we point the reader to (Nisan et al. 2007) sources that are very popular. We present the results for two for the main results of the area). The study of the ineffi- datasets below, and give additional simulation results in our ciency of further equilibrium concepts naturally followed, full version (Gollapudi, Kollias, and Panigrahi 2018). All including the strong Nash equilibrium (Aumann 1959; An- the simulation results exhibit similar trends, where SHAP- delman, Feldman, and Mansour 2009; Bachrach et al. 2013; LEY improves significantly over EQUAL with respect to the Roughgarden and Schrijvers 2014). parameters described above. Vetta (Vetta 2002) considered a broad class of utility Skewed Agent Resource Ownage. For our first simulation, games and gave an upper bound of 2 on the price of an- we assume that the 20 resource type frequencies are sym- archy. The interpretation of the main assumption of (Vetta metric (i.e., we expect approximately the same number of 2002) in our setting is that the social welfare function in agents will own each one). On the contrary, we assume that each sharing group is submodular. While this is a reasonable the agent resource ownage, defined as the number of re- assumption in many settings, this does not hold in the appli- source types the agent owns, follows a power law distribu- cations we study, where submodular utility functions do not tion. The plots in Figures 1a, 1b, and 1c show how our three properly model realistic scenarios and lead to agents form- metrics vary as the average ownage level changes (i.e., as ing singleton groups. Following up on (Vetta 2002), other the degree of the power law changes). more specific applications have been considered in (Marden Performance-wise, we see in Figure 1a that SHAPLEY im- and Wierman 2013; Marden and Roughgarden 2014) as also proves the total utility by approximately 10% for all agent a setting with incomplete information (Bachrach, Syrgkanis, ownage levels. We can also see (from Figure 1b) that agents and Vojnovic 2013). consistently participate in sharing groups when SHAPLEY is Coalitionally robust resource sharing games can be placed applied. In the proof of Theorem 12, we showed that when in the context of hedonic games, a term coined by Dreze the number of groups is large, the BPOA of SHAPLEY is up- and Greenberg (Dreze and Greenberg 1980). The earli- per bounded by 3. On the other hand, in Theorem 5, we est and most well-known result in the area is the one on showed that the POA of EQUAL is k = 5. The improvements stable marriages by Gale and Shapley (Gale and Shapley from SHAPLEY indeed verify our theoretical analysis in this 1962). Further works include (Aumann and Dreze 1974; setting. Apt et al. 2014; Banerjee, Konishi, and Sonmez 2001; Figure 1b shows that EQUAL can have unfilled groups as Bogomolnaia and Jackson 2002; Chalkiadakis et al. 2009; the average agent ownage level drops, since there are more Feldman, Lewin-Eytan, and Naor 2012; Greenberg and We- agents who cannot access any resource type. Since the other ber 1993; Hajdukova 2006; Hart and Kurz 1983; Immorlica, agents are clustered in groups, they are left out and have no Markakis, and Piliouras 2010; Yi 1997). incentive to join each other. For the same reason, EQUAL The Shapley value, upon which our SHAPLEY method performs poorly with respect to minimum utility, as we see is based, is defined in (Shapley 1953). The Shapley value

2010 has been used in utility games (Marden and Wierman 2013; Chalkiadakis, G.; Elkind, E.; Polukarov, M.; and Jennings, N. R. Marden and Roughgarden 2014) and in network routing 2009. The price of democracy in coalition formation. In 8th Inter- games (Kollias and Roughgarden 2011). The strong price of national Joint Conference on Autonomous Agents and Multiagent anarchy of games induced by the Shapley value in network Systems (AAMAS 2009), 401–408. cost-sharing games is studied in (Roughgarden and Schri- Davis, B. G. 2009. Tools for Teaching. jvers 2014). Dreze, J. H., and Greenberg, J. 1980. Hedonic coalitions: Optimal- ity and stability. Econometrica 48(4):987–1003. Conclusion Feldman, M.; Lewin-Eytan, L.; and Naor, J. 2012. Hedonic clus- tering games. In 24th ACM Symposium on Parallelism in Algo- In this paper, we assumed the central authority’s perspec- rithms and Architectures, SPAA ’12, Pittsburgh, PA, USA, June 25- tive in sharing economy settings where agents organically 27, 2012, 267–276. form sharing groups to maximize their individual utility. We Gairing, M. 2009. Covering games: Approximation through non- showed that a Nash equilibrium (NE) is always guaranteed cooperation. In WINE, 184–195. to exist, but all natural utility sharing methods are inefficient Gale, D., and Shapley, L. S. 1962. College admissions and the sta- for the corresponding price of anarchy (POA) metric. In con- bility of marriage. The American Mathematical Monthly 69(1):9– trast, a Shapley utility sharing method (SHAPLEY) is highly 15. efficient for the strong POA metric, and significantly out- Gollapudi, S.; Kollias, K.; and Panigrahi, D. 2018. 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